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arxiv: 2409.09862 · v4 · submitted 2024-09-15 · 🧬 q-bio.NC

Towards a Quantitative Theory of Digraph-Based Complexes and its Applications in Brain Network Analysis

Heitor Baldo This is my paper

Pith reviewed 2026-05-23 20:39 UTC · model grok-4.3

classification 🧬 q-bio.NC
keywords digraph-based complexesdirected clique complexespath complexeshigher-order topologybrain network analysisseizure biomarkersiPDCepilepsy
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The pith

Higher-order structures in directed brain networks from iPDC provide new biomarkers for seizure dynamics and focus laterality.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops quantitative methods for digraph-based complexes such as path complexes and directed clique complexes, including measures of directed higher-order connectivities between cliques. These methods are then applied to digraphs obtained from iPDC estimates on EEG recordings of patients with left temporal lobe epilepsy, across delta, theta, and alpha bands. The central effort is to track how these higher-order topologies shift from pre-ictal to ictal to post-ictal phases in each hemisphere and to test whether the resulting measures improve on standard graph-theoretic biomarkers for seizure dynamics and seizure-focus laterality. A sympathetic reader would care because the work claims that pairwise connections alone miss directed higher-order features that better reflect the biological reorganization during seizures.

Core claim

The paper claims that characterization and similarity measures defined on path complexes and directed clique complexes, together with directed higher-order connectivities, applied to iPDC-derived digraphs, detect systematic changes in higher-order topology across seizure phases and hemispheres that are not fully captured by usual graph measures, thereby supplying candidate biomarkers for seizure dynamics and laterality of the seizure focus.

What carries the argument

Directed clique complexes and path complexes constructed from digraphs, plus the directed higher-order connectivities that relate the cliques to one another, which serve as the quantitative objects whose characterization and similarity measures are computed.

If this is right

  • Higher-order topology changes measurably between pre-ictal, ictal, and post-ictal phases in each frequency band and hemisphere.
  • Directed higher-order connectivities supply additional quantitative descriptors beyond those available from pairwise graphs.
  • These descriptors can be compared across patients to assess laterality of the seizure focus.
  • The same quantitative apparatus formalizes a general theory for measuring properties of digraph-based complexes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same complex-based measures could be tested on other directed connectivity estimators to check whether the observed phase changes are robust to the choice of estimator.
  • If the biomarkers prove stable, they could be examined for real-time detection of seizure onset in clinical monitoring settings.
  • The framework might be applied to directed networks outside epilepsy, such as those recorded during cognitive tasks, to see whether higher-order directed features appear more broadly.

Load-bearing premise

The higher-order topology extracted from iPDC digraphs records genuine seizure-related biological changes rather than artifacts of the connectivity estimator or the complex construction.

What would settle it

An independent EEG dataset in which the higher-order measures show no additional predictive value for phase transitions or focus laterality once standard pairwise graph metrics are already included would falsify the claim of improvement.

Figures

Figures reproduced from arXiv: 2409.09862 by Heitor Baldo.

Figure 2.1
Figure 2.1. Figure 2.1: Examples of (k + 1)-cliques, for k = 0, 1, 2, 3, 4. The term “clique” originated in the study of social networks to denote the formation of a group of two or more people if the condition of being mutual friends is satisfied [174], and later was adopted to denote a complete induced subgraph. The process of clique enumeration (finding and listing all cliques in a graph) is very useful and widely used in ne… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Examples of directed and weighted graphs (the weights [PITH_FULL_IMAGE:figures/full_fig_p034_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Networks obtained through the NetLogo software [ [PITH_FULL_IMAGE:figures/full_fig_p056_2_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Examples of geometric simplices. Definition 3.1.3. The vertex scheme of a geometric simplicial complex is the ASC built out of the sets of vertices of its geometric simplices. On the other hand, the geometric realization of an ASC ∆ is the geometric simplicial complex whose vertex scheme is isomorphic to ∆. Every finite ASC has a geometric realization on an Euclidean space, as stated by the geometric rea… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Different orientations of a 2-simplex. In summary, an ordered n-simplex is merely an n-simplex with a total ordering in its vertices. Therefore, we can formulate an analogous definition of the definition of ASC where our simplices are ordered simplices, i.e. the definition of abstract ordered simplicial complex or abstract directed simplicial complex [218]. Definition 3.1.4. An abstract directed simplici… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Examples of directed (k + 1)-cliques, for k = 0, 1, 2, 3, 4. Definition 3.1.9. Given a digraph G = (V, E), its directed flag complex, denoted by dFl(G), is the abstract directed simplicial complex whose directed k-simplices span directed (k + 1)-cliques of G, i.e. for every [v0, ..., vk] ∈ dFl(G), we have vi ∈ V , ∀i, and (vi , vj ) ∈ E, ∀i < j. The directed simplices of a directed flag complex are not u… view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: A digraph G with a double-edge and its directed flag complex dFl(G). Moreover, note that directed cycles are not considered directed cliques since they do not have a source and a sink. We say that the flag complex associated with the underlying undirected graph of a digraph is its underlying flag complex. Example 3.1.6 [PITH_FULL_IMAGE:figures/full_fig_p065_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: A digraph G along with its underlying flag complex Fl(G), and its directed flag complex dFl(G). Remark 3.1.2. Directed cliques are combinatorial objects, thus, for instance, given a set of three vertices, we can build six directed 3-cliques by changing the directions of their edges in a proper way. They are all isomorphic because we can transform one into the other just by reordering their vertices and, … view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: A weighted digraph and its pre-metric Dowker complex for [PITH_FULL_IMAGE:figures/full_fig_p069_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: There is a 1-dimensional hole associated with a 3-cycle, but there is no [PITH_FULL_IMAGE:figures/full_fig_p071_3_7.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Example of filtration. There are several different ways to create different filtrations for a (unweighted or weighted) simplicial complex [83, 219]. One way is by considering it vertices in a metric space, and then producing simplices (and thus subcomplexes) by gradually increasing the threshold on the distance between the vertices, similarly as specified in the Definition 3.1.13 of the Vietoris-Rips com… view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: The persistence diagram, persistence barcodes, and Betti curves correspond [PITH_FULL_IMAGE:figures/full_fig_p076_3_9.png] view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Digraph G. B1 =       [01] [02] [12] [13] [23] [0] 1 1 0 0 0 [1] −1 0 1 1 0 [2] 0 −1 −1 0 1 [3] 0 0 0 −1 −1       , B2 =         [012] [123] [01] 1 0 [02] −1 0 [12] 1 1 [13] 0 −1 [23] 0 1         . Note that Bn = 0 for all n ≥ 3. Thus, by the formula 3.27, the Hodge n-Laplacians associated with X are: [L0] = B1B T 1 =      2 −1 −1 0 −1 3 −1 −1 −1 −1 3 −1 0 −1 −1 2    … view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: A digraph and its path complex. Note that every p-path of a path complex is allowed. Restricting the operator ∂p on the subspace Ap ⊆ Λp, p ≥ 0, we can have ∂pAp ̸⊂ Ap−1, but we are interested in the case in which the inclusion occurs, thus we define the following subspace of Ap. Definition 3.2.6. Given a digraph G = (V, E), consider the following subspace of Ap(V ), p ≥ 0: Ωp = Ωp(G) := {u ∈ Ap : ∂pu ∈… view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Examples of digraphs containing ∂-invariant 2-paths. Moreover, Grigor’yan et al. [127] proved that the elements of Ω2 are linear combi￾nations of triangles, squares, and double-edges. Snakes and ∂-Invariant Directed Quasi-Cliques The presence or absence of ∂-invariant paths in a digraph can characterize some of its topological properties, since these paths are related to the path homology of the digraph… view at source ↗
Figure 3.13
Figure 3.13. Figure 3.13: Examples of p-snakes for p = 2, 3, 4. In the next definition we present a directed version of the definition of γ-quasi-clique (Definition 2.1.19). Definition 3.2.8. Given a digraph G = (V, E), a subdigraph H = (V ′ , E′ ) ⊆ G, with |V ′ | = m, is called a directed γ-quasi-clique (or γ-DQC), for a parameter 0 < γ ≤ 1, if degtot H (v) ≥ γ(m − 1), for all v ∈ V ′ . 74 [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: The (ui , γ)-DQCs associated with elementary ∂-invariant p-paths, with γ = 1/p. (a) u1 = e012. (b)u2 = e0123. (c) u3 = e01234 (d) u4 = e012345. 75 [PITH_FULL_IMAGE:figures/full_fig_p083_3_14.png] view at source ↗
Figure 3.15
Figure 3.15. Figure 3.15: Examples of (u, γ)-DQCs associated with ∂-invariant 3-paths. (a) u = e0123 − e01′23. (b) u2 = e0123 − e012′3. (c) u3 = e0123 − e012′3 + e01′2 ′3. 3.2.2 Path Homology In this part, we extend the concept of simplicial homology introduced for simplicial complexes in Subsection 3.1.4 to path complexes. We begin by pointing out that the boundary operators (3.28) restricted to the spaces Ω• satisfy the same p… view at source ↗
Figure 3.16
Figure 3.16. Figure 3.16: A simplicial complex. 3.3.2 Directed Q-Analysis and Directed Higher-Order Adja￾cencies As we have seen in the previous section, Atkin’s Q-Analysis defines q-connectivity be￾tween two simplices based solely on the face shared by them and does not say anything about the directionality of this connection. Recently, however, H. Riihimaki [220] in￾troduced a formalism to treat the q-connectivity between dire… view at source ↗
Figure 3.17
Figure 3.17. Figure 3.17: A directed flag complex. Lower, Upper, and General Adjacencies In what follows, we extend the definitions of lower, upper, and general adjacencies as exposed in [233] to directed simplices. Definition 3.3.13. For two directed simplices σ (n) , τ (m) ∈ dFl(G) and for 0 ≤ q ≤ min(n, m), we have the following definitions: 1. σ (n) is lower (•)-q-adjacent to τ (m) , where • ∈ {−, +, ±}, if and only if σ (n)… view at source ↗
Figure 3.18
Figure 3.18. Figure 3.18: Graphical representation of the maximal directed simplices for each level [PITH_FULL_IMAGE:figures/full_fig_p097_3_18.png] view at source ↗
Figure 3.19
Figure 3.19. Figure 3.19: A directed flag complex and its respective maximal [PITH_FULL_IMAGE:figures/full_fig_p098_3_19.png] view at source ↗
Figure 3.20
Figure 3.20. Figure 3.20: A weighted digraph together with its weighted directed flag complex and [PITH_FULL_IMAGE:figures/full_fig_p100_3_20.png] view at source ↗
Figure 3.21
Figure 3.21. Figure 3.21: A directed flag complex. Lower, Upper, and General Degrees Based on the previous definitions of q-stars, in this last part we extend the definitions of lower, upper, and general degrees to directed simplices. Definition 3.3.31. Given a directed simplex σ (n) ∈ dFl(G), for 0 ≤ q ≤ n and for • ∈ {−, +, ±}, we have the following definitions: 1. The lower (•)-q-degree of σ (n) is defined by deg• Lq (σ (n) )… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: A directed flag complex [PITH_FULL_IMAGE:figures/full_fig_p111_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: q-Forman-Ricci curvatures for q-arcs and in- and out-q-Forman-Ricci cur￾vatures for nodes in the q-digraphs associated with the directed flag complex shown on the left side, for q = 0, 1. 108 [PITH_FULL_IMAGE:figures/full_fig_p116_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Random digraphs and their respective q-digraphs (q = 0, 1, 2) [PITH_FULL_IMAGE:figures/full_fig_p125_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Means and standard deviations of the four simplicial measures computed [PITH_FULL_IMAGE:figures/full_fig_p127_4_4.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Graphical representation of a neuron and an action potential. [PITH_FULL_IMAGE:figures/full_fig_p131_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: International 10-20 electrode system, with 21 electrodes. [PITH_FULL_IMAGE:figures/full_fig_p133_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Examples of iDTF and iPDC networks. Dashed lines represent weak con [PITH_FULL_IMAGE:figures/full_fig_p145_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Structural and functional connectivity networks corresponding to patient 1 [PITH_FULL_IMAGE:figures/full_fig_p148_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: G-connectivity (iPDC) and G-influentiablility (iDTF) digraphs, along with [PITH_FULL_IMAGE:figures/full_fig_p149_5_5.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: EEG data preprocessing workflow [PITH_FULL_IMAGE:figures/full_fig_p163_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Configuration of the 26 selected electrodes (represented in black), according [PITH_FULL_IMAGE:figures/full_fig_p163_7_2.png] view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: Example of preprocessed signal corresponding to patient PN01 (crisis 1). [PITH_FULL_IMAGE:figures/full_fig_p164_7_3.png] view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: iPDC networks (arcs thicknesses are proportional to the weights) of patient [PITH_FULL_IMAGE:figures/full_fig_p167_7_4.png] view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: iPDC networks (arcs thicknesses are proportional to the weights) of patient [PITH_FULL_IMAGE:figures/full_fig_p168_7_5.png] view at source ↗
Figure 7.6
Figure 7.6. Figure 7.6: summarizes the analysis work flow, including the preprocessing step [PITH_FULL_IMAGE:figures/full_fig_p169_7_6.png] view at source ↗
read the original abstract

In this work, we developed new mathematical methods for analyzing network topology and applied these methods to the analysis of brain networks. More specifically, we rigorously developed quantitative methods based on complexes constructed from digraphs (digraph-based complexes), such as path complexes and directed clique complexes (alternatively, we refer to these complexes as "higher-order structures," or "higher-order topologies," or "simplicial structures"), and, in the case of directed clique complexes, also methods based on the interrelations between the directed cliques, what we called "directed higher-order connectivities." This new quantitative theory for digraph-based complexes can be seen as a step towards the formalization of a "quantitative simplicial theory." Subsequently, we used these new methods, such as characterization measures and similarity measures for digraph-based complexes, to analyze the topology of digraphs derived from brain connectivity estimators, specifically the estimator known as information partial directed coherence (iPDC), which is a multivariate estimator that can be considered a representation of Granger causality in the frequency-domain, particularly estimated from electroencephalography (EEG) data from patients diagnosed with left temporal lobe epilepsy, in the delta, theta and alpha frequency bands, to try to find new biomarkers based on the higher-order structures and connectivities of these digraphs. In particular, we attempted to answer the following questions: How does the higher-order topology of the brain network change from the pre-ictal to the ictal phase, from the ictal to the post-ictal phase, at each frequency band and in each cerebral hemisphere? Does the analysis of higher-order structures provide new and better biomarkers for seizure dynamics and also for the laterality of the seizure focus than the usual graph theoretical analyses?

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The paper develops new quantitative methods for digraph-based complexes (path complexes, directed clique complexes) and directed higher-order connectivities, then applies them to iPDC-derived digraphs from EEG recordings in left temporal lobe epilepsy patients. It seeks to characterize changes in higher-order topology across pre-ictal/ictal/post-ictal phases in delta/theta/alpha bands and to establish that these structures yield superior biomarkers for seizure dynamics and seizure-focus laterality relative to standard graph-theoretic measures.

Significance. If the central claims were supported by derivations and empirical validation, the work would supply a formal quantitative framework for directed higher-order topology and demonstrate its utility for epilepsy biomarker discovery beyond pairwise network metrics.

major comments (3)
  1. [Abstract] Abstract: the manuscript asserts that 'rigorously developed' quantitative methods based on digraph complexes were constructed and applied to answer concrete questions about phase-dependent topology changes and biomarker superiority, yet supplies no equations, derivations, characterization measures, similarity measures, or statistical results to substantiate any of these claims.
  2. [Abstract] Abstract: the claim that directed clique/path complexes and directed higher-order connectivities furnish 'new and better biomarkers' for seizure dynamics and focus laterality than usual graph-theoretic analyses requires evidence that the higher-order features are not redundant with pairwise measures (edge density, clustering, centrality); no ablation, feature-importance, or predictive-performance comparison is presented.
  3. [Abstract] Abstract: the weakest assumption—that changes observed in the higher-order structures across phases reflect genuine seizure-related dynamics rather than iPDC estimation artifacts—is left untested; the text contains no validation against surrogate data, robustness checks, or comparison with conventional iPDC-derived graph metrics.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed comments on our manuscript. We address each major comment below, clarifying the scope of our claims and indicating where revisions can strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the manuscript asserts that 'rigorously developed' quantitative methods based on digraph complexes were constructed and applied to answer concrete questions about phase-dependent topology changes and biomarker superiority, yet supplies no equations, derivations, characterization measures, similarity measures, or statistical results to substantiate any of these claims.

    Authors: The abstract provides a high-level summary of the contributions, as is conventional. The main text develops the quantitative methods for digraph-based complexes (path complexes and directed clique complexes) with explicit mathematical definitions, derivations of characterization measures (including dimension-specific simplex counts and adapted topological invariants), similarity measures between complexes, and directed higher-order connectivities. The results section reports statistical comparisons of these quantities across pre-ictal, ictal, and post-ictal phases in the specified frequency bands. If the referee finds the derivations insufficiently detailed or the abstract misleading in its phrasing, we will expand the relevant sections and adjust the abstract wording in revision. revision: partial

  2. Referee: [Abstract] Abstract: the claim that directed clique/path complexes and directed higher-order connectivities furnish 'new and better biomarkers' for seizure dynamics and focus laterality than usual graph-theoretic analyses requires evidence that the higher-order features are not redundant with pairwise measures (edge density, clustering, centrality); no ablation, feature-importance, or predictive-performance comparison is presented.

    Authors: The manuscript does not assert that the higher-order structures furnish superior biomarkers. It states that the methods were applied 'to try to find new biomarkers' and explicitly poses the question 'Does the analysis of higher-order structures provide new and better biomarkers... than the usual graph theoretical analyses?' as one of the motivating questions. The results include direct comparisons to selected standard graph metrics, but we acknowledge that a systematic ablation study, feature-importance ranking, or predictive-performance evaluation to assess redundancy is absent. We will add these analyses in a revised version. revision: yes

  3. Referee: [Abstract] Abstract: the weakest assumption—that changes observed in the higher-order structures across phases reflect genuine seizure-related dynamics rather than iPDC estimation artifacts—is left untested; the text contains no validation against surrogate data, robustness checks, or comparison with conventional iPDC-derived graph metrics.

    Authors: The manuscript does perform comparisons between the higher-order measures and conventional iPDC-derived graph metrics in the results. However, we agree that explicit validation against surrogate data (e.g., phase-randomized or amplitude-adjusted surrogates) and additional robustness checks would be necessary to strengthen the claim that observed changes reflect genuine dynamics rather than estimation artifacts. We will incorporate surrogate analyses and expanded robustness checks in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain self-contained with no reductions to inputs

full rationale

The abstract and provided text describe the construction of new quantitative methods for digraph-based complexes (path complexes, directed clique complexes, directed higher-order connectivities) and their application to iPDC-derived brain networks for biomarker discovery. No equations, parameter fittings, self-citations, or ansatzes are exhibited that reduce a claimed prediction or result to the input data or prior definitions by construction. The work positions itself as developing independent formal tools toward a quantitative simplicial theory and testing empirical questions on seizure dynamics; absent any load-bearing self-referential steps in the given material, the chain is self-contained against external benchmarks. The orthogonality of higher-order features to pairwise metrics is an empirical claim, not a definitional circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the methods are described at a high level without mathematical detail.

pith-pipeline@v0.9.0 · 5853 in / 1137 out tokens · 30490 ms · 2026-05-23T20:39:55.918959+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We rigorously developed quantitative methods based on complexes constructed from digraphs (digraph-based complexes), such as path complexes and directed clique complexes... and... directed higher-order connectivities.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    all simplicial characterization measures... showed statistically significant increases... from the pre-ictal phase to the ictal phase... no statistically significant changes... from the ictal to the post-ictal phase

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The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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